A Unified Analysis on the Subgradient Upper Bounds for the Subgradient Methods Minimizing Composite Nonconvex, Nonsmooth and Non-Lipschitz Functions
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911392303415296 |
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| author | Zhu, Daoli Zhao, Lei Zhang, Shuzhong |
| author_facet | Zhu, Daoli Zhao, Lei Zhang, Shuzhong |
| contents | This paper presents a unified analysis for the proximal subgradient method (Prox-SubGrad) type approach to minimize an overall objective of $f(x)+r(x)$, subject to convex constraints, where both $f$ and $r$ are weakly convex, nonsmooth, and non-Lipschitz. Leveraging on the properties of the Moreau envelope of weakly convex functions, we are able to relate error-bound conditions, the growth conditions of the subgradients of the objective, and the behavior of the proximal subgradient iterates on some remarkably broad classes of objective functions. Various existing as well as new bounding conditions are studied, leading to novel iteration complexity results. The terrain of our exploration expands to stochastic proximal subgradient algorithms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_16362 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Unified Analysis on the Subgradient Upper Bounds for the Subgradient Methods Minimizing Composite Nonconvex, Nonsmooth and Non-Lipschitz Functions Zhu, Daoli Zhao, Lei Zhang, Shuzhong Optimization and Control Machine Learning 90C33, 65K15, 90C46, 49K35, 90C26 This paper presents a unified analysis for the proximal subgradient method (Prox-SubGrad) type approach to minimize an overall objective of $f(x)+r(x)$, subject to convex constraints, where both $f$ and $r$ are weakly convex, nonsmooth, and non-Lipschitz. Leveraging on the properties of the Moreau envelope of weakly convex functions, we are able to relate error-bound conditions, the growth conditions of the subgradients of the objective, and the behavior of the proximal subgradient iterates on some remarkably broad classes of objective functions. Various existing as well as new bounding conditions are studied, leading to novel iteration complexity results. The terrain of our exploration expands to stochastic proximal subgradient algorithms. |
| title | A Unified Analysis on the Subgradient Upper Bounds for the Subgradient Methods Minimizing Composite Nonconvex, Nonsmooth and Non-Lipschitz Functions |
| topic | Optimization and Control Machine Learning 90C33, 65K15, 90C46, 49K35, 90C26 |
| url | https://arxiv.org/abs/2308.16362 |